Definition

Definition

Remark

This means that 0∈𝒞0 \in \mathcal{C} is a zero object precisely if for every other object AA there is a unique morphismA→0A \to 0 to the zero object as well as a unique morphism 0→A0 \to A from the zero object.

Remark

There is also a notion of zero object in algebra which does not always coincide with the category-theoretic terminology. For example the zero ring{0}\{0\} is not an initial object in the category of unital rings (this is instead the integersℤ\mathbb{Z}); but it is the terminal object. However, the zero ring is the zero object in the category of nonunital rings (although it happens to be unital).

Proof

Write *∈Set** \in Set_* for the singleton pointed set. Suppose tt is terminal. Then C(x,t)=*C(x,t) = * for all xx and so in particular C(t,t)=*C(t,t) = * and hence the identity morphism on tt is the basepoint in the pointed hom-set. By the axioms of a category, every morphism f:t→xf : t \to x is equal to the composite

f:t→Idt→fx.
f : t \stackrel{Id}{\to} t \stackrel{f}{\to} x
\,.

By the axioms of an (Set*,∧)(Set_*, \wedge)-enriched category, since IdtId_{t} is the basepoint in C(t,t)C(t,t), also this composite is the basepoint in C(t,x)C(t,x) and is hence the zero morphism. So C(t,x)=*C(t,x) = * for all xx and therefore tt is also an initial object.

Analogously from assuming tt to be initial it follows that it is also terminal.